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Persistence of SPT-Legality from Channelized Feasibility in a Multiscale Hilbert/Cell Model
When does being spread out at every scale survive the dynamics? Exact conditions in a finite model.
In plain words
Multiscale arguments often check that a state is spread out at fine scales, not piled up in one small region, and then assume the dynamics keeps it that way. This paper asks when that assumption is justified. The model is finite and linear. Space is split into depth bands, and each band into cells. A number measures concentration in each band. It is 1 when energy is spread evenly over the cells, and it equals the number of cells when everything sits in one cell. A state is called SPT legal if this number stays below a fixed threshold at every depth.
The paper does not try to derive spreading from the dynamics. Instead, the allowed band states are built from a fixed set of channels. If each channel is spread out, the channels are stably independent, and their number is controlled, every allowed state obeys an explicit bound, and the channel count factor in it cannot be removed. If the dynamics keeps allowed states allowed, the bound holds for all time; being legal at the start is not enough. If the dynamics only keeps the disallowed part small in every cell, a slightly weaker bound still holds.
Each hypothesis has an explicit counterexample when it is dropped: a single localized channel, nearly dependent channels, an unchecked channel count, or leakage measured against the whole state instead of the band. Even a channel count that grows only linearly with depth allows unbounded concentration. Conservation of energy alone places no limit on concentration beyond the trivial one. The three bounds are machine checked in Lean 4 with Mathlib, and nine numerical experiments illustrate them.
What it shows
- A static concentration bound from spread out, stable channels, sharp in the channel count.
- Persistence for all time when the dynamics preserves the allowed set; legality at the start does not suffice.
- A robust version when only a small leak per cell is allowed.
- A counterexample for every weakened hypothesis, and energy conservation alone is not enough.
What it does not claim
The results concern finite dimensional linear models. They make no claim about smoothing or regularity for partial differential equations, and they do not say how a suitable set of channels would arise from a given system.
Cite
Tsiokos, I. (2026). Persistence of SPT-Legality from Channelized Feasibility in a Multiscale Hilbert/Cell Model. Zenodo. https://doi.org/10.5281/zenodo.23118447